Quantum dots on the nuclear scale! Quantum dots are quantum systems that are confined by definition on the nano scale. Why didn't people study similar systems on a much smaller scale, something as small as the dimension of the nucleus? Is it practically difficult? or useless?
R.
 A: normal matter structure is entirely constructed from the electronic bindings, so it is in the realm of the possible to engineer how the atoms are binded together exactly and this is the aim of lower-level nanotechnology. However, it is with current technogy and physics, impossible to create complex structures at lower scales (i.e: nuclear scale). And i don't think that is something that is in principle possible unless some dramatic breakthrough occurs in how we obtain larger-scale nuclear matter, which from the experimental limitations that there are to obtain high Z nucleus that might probe the regions of higher islands of stability, one can safely infer that this is 
a very challenging problem in of itself
Update: this is not entirely related, but it shows an example of how assumptions as the one i've made above about manipulation of small scales being out of engineering reach can be twisted: this slide about non-homogeneous diffraction crystals shows how to do something that most physicists have thought for long to be essentially impossible; X-ray and gamma-ray optics
A: Depends on your definition of a quantum dot. Usually they are understood to be a collection of excitons (bound states of electrons and holes) confined to a small volume. But the smaller the volume, the higher the energy of the system and eventually you would destroy the exciton bonds (besides, there would be a technical problem of how to maintain such a localized potential well).
So I suppose by quantum dot you mean arbitrary quantum system that is very localized. In this case, nucleus itself should satisfy you well.
A: The term quantum dot is, as you said, used to describe the need to treat a system on the nanoscale quantum-mechanically. This is important for example in nanophotonics where e.g. a coupling of a large system to such a quantum mechanical one leads to interesting physics of nanoscale devices. See e.g. Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity (Yoshie et al, Nature, 2004) or Exciton-Photon Strong-Coupling Regime for a Single Quantum Dot Embedded in a Microcavity (Peter et al, PRL, 2005).
But bringing this notion back to where it came from - quantum systems which can only be described by quantum mechanics - does not make sense. 
Greets
A: People did study similar systems on a much smaller scale, and you probably know about those studies. Hint: another name for “quantum dot” is “artificial atom”.
